We study the geometry of simplicial cochains on Vietoris–Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper-bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

This is joint work with Darrick Lee: https://arxiv.org/pdf/2608.11137

Hyperquiver representations are  generalisations of quiver representations. They are directed hypergraphs with vector spaces assigned to vertices, and for every directed hyperedge, an assignment of a linear map from the tensor product of the source vector spaces to the tensor product of the target vector spaces. We study the moduli of hyperquiver representations, where we use affine GIT to describe the (in)stability of such objects under the action of a product of general linear groups. Our main result introduces the Harder-Narasimhan filtration of a hyperquiver representation. We also derive a formula for the weights of this unique filtration, which specialise to the Harder-Narasimhan slopes of a quiver representation.

Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH: given a persistence module, what can we say about the data that generated it? To study the geometry of the fibers of MPH, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications that are preserved by the map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.

Kesten's theorem characterises amenability in terms of spectral radii: a symmetric random walk on a group G has spectral radius 1 if and only if it generates an amenable subgroup.

We prove a quantitative "1%" version of this result for G a free group, or a fully residually free group: for every ε > 0, there is δ > 0 such that for every probability measure μ on G with convolution operator norm at least ε, there is a co-set of an amenable subgroup of G of mass > δ under μ.

This answers a question of Emmanuel Breuillard. We believe the same result to hold in a broader setting of linear groups, with δ depending on the dimension. The proof uses uniform exactness and a generalisation of the Tits alternative.

Joint work with Emmanuel Breuillard.
 

TBA